Hodge diamonds of the Landau--Ginzburg orbifolds

TL;DR

This paper studies the Hodge structure of Landau-Ginzburg orbifolds, showing that their B-model state space forms a diamond-shaped bigraded structure with specific isomorphisms, extending understanding of singularity invariants.

Contribution

It proves that the nonvanishing bigraded pieces of the B-model state space form a diamond and identifies key entries and isomorphisms, generalizing previous results to non-abelian groups.

Findings

The B-model state space forms a diamond-shaped bigrading.

Topmost, bottommost, leftmost, rightmost entries are one-dimensional.

The diamond exhibits essential horizontal and vertical isomorphisms.

Abstract

Consider the pairs $(f,G)$ with $f = f(x_1,\dots,x_N)$ being a polynomial defining a quasihomogeneous singularity and $G$ being a subgroup of ${\rm SL}(N,\mathbb{C})$, preserving $f$. In particular, $G$ is not necessary abelian. Assume further that $G$ contains the grading operator $j_f$ and $f$ satisfies the Calabi-Yau condition. We prove that the nonvanishing bigraded pieces of the B-model state space of $(f,G)$ form a diamond. We identify its topmost, bottommost, leftmost and rightmost entries as one-dimensional and show that this diamond enjoys the essential horizontal and vertical isomorphisms.